The slope determines if the function is an
increasing linear function , a
decreasing linear function , or a constant function.
$f(x)=mx+b\text{\hspace{0.17em}}$ is an increasing function if
$\text{\hspace{0.17em}}m>0.$
$f(x)=mx+b\text{\hspace{0.17em}}$ is a decreasing function if
$\text{\hspace{0.17em}}m<0.$
$f(x)=mx+b\text{\hspace{0.17em}}$ is a constant function if
$\text{\hspace{0.17em}}m=0.$
Deciding whether a function is increasing, decreasing, or constant
Some recent studies suggest that a teenager sends an average of 60 texts per day
http://www.cbsnews.com/8301-501465_162-57400228-501465/teens-are-sending-60-texts-a-day-study-says/ . For each of the following scenarios, find the linear function that describes the relationship between the input value and the output value. Then, determine whether the graph of the function is increasing, decreasing, or constant.
The total number of texts a teen sends is considered a function of time in days. The input is the number of days, and output is the total number of texts sent.
A teen has a limit of 500 texts per month in his or her data plan. The input is the number of days, and output is the total number of texts remaining for the month.
A teen has an unlimited number of texts in his or her data plan for a cost of $50 per month. The input is the number of days, and output is the total cost of texting each month.
Analyze each function.
The function can be represented as
$\text{\hspace{0.17em}}f(x)=60x\text{\hspace{0.17em}}$ where
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is the number of days. The slope, 60, is positive so the function is increasing. This makes sense because the total number of texts increases with each day.
The function can be represented as
$\text{\hspace{0.17em}}f(x)=500-60x\text{\hspace{0.17em}}$ where
$\text{\hspace{0.17em}}x$ is the number of days. In this case, the slope is negative so the function is decreasing. This makes sense because the number of texts remaining decreases each day and this function represents the number of texts remaining in the data plan after
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ days.
The cost function can be represented as
$\text{\hspace{0.17em}}f(x)=50\text{\hspace{0.17em}}$ because the number of days does not affect the total cost. The slope is 0 so the function is constant.
In the examples we have seen so far, the slope was provided to us. However, we often need to calculate the slope given input and output values. Recall that given two values for the input,
$\text{\hspace{0.17em}}{x}_{1}$ and
$\text{\hspace{0.17em}}{x}_{2},$ and two corresponding values for the output,
$\text{\hspace{0.17em}}{y}_{1}\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}{y}_{2}\text{\hspace{0.17em}}$ —which can be represented by a set of points,
$\text{\hspace{0.17em}}\left({x}_{1}\text{,}{y}_{1}\right)\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}\left({x}_{2}\text{,}{y}_{2}\right)$ —we can calculate the slope
$m.$
Note that in function notation we can obtain two corresponding values for the output
$\text{\hspace{0.17em}}{y}_{1}\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}{y}_{2}\text{\hspace{0.17em}}$ for the function
$\text{\hspace{0.17em}}f,$$\text{\hspace{0.17em}}{y}_{1}=f\left({x}_{1}\right)\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}{y}_{2}=f\left({x}_{2}\right),$ so we could equivalently write
[link] indicates how the slope of the line between the points,
$\text{\hspace{0.17em}}\left({x}_{1},{y}_{1}\right)\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}\left({x}_{2},{y}_{2}\right),\text{\hspace{0.17em}}$ is calculated. Recall that the slope measures steepness, or slant. The greater the absolute value of the slope, the steeper the slant is.
Questions & Answers
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5) and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.